On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

摘要

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ≥ n, given by F 1(z) = (0, I n;m (z)) with the isometric constant equal to (m + 1)/(n + 1) and F 2(z) = (I n;m (z), I n;m (z)) with the isometric constant equal to 2(m + 1)/(n + 1). Here ${I_{n;m}:mathbb{C}^n longrightarrow mathbb{C}^m}$ is the canonical embedding. We prove that when m 2n, these are the only holomorphic isometric embeddings up to unitary transformations.